The homology of string algebras I

Abstract

We show that string algebras are `homologically tame' in the following sense: First, the syzygies of arbitrary representations of a finite dimensional string algebra $\Lambda$ are direct sums of cyclic representations, and the left finitistic dimensions, both little and big, of $\Lambda$ can be computed from a finite set of cyclic left ideals contained in the Jacobson radical. Second, our main result shows that the functorial finiteness status of the full subcategory $\Cal P$ consisting of the finitely generated left $\Lambda$-modules of finite projective dimension is completely determined by a finite number of, possibly infinite dimensional, string modules -- one for each simple $\Lambda$-module -- which are algorithmically constructible from quiver and relations of $\Lambda$. Namely, $\Cal P$ is contravariantly finite in $\Lambda$-mod precisely when all of these string modules are finite dimensional, in which case they coincide with the minimal $\Cal P$-approximations of the corresponding simple modules. Yet, even when $\Cal P$ fails to be contravariantly finite, these `characteristic' string modules encode, in an accessible format, all desirable homological information about $\Lambda$-mod.